Searching for rational points on varieties over global fields

When

Start: 02/26/2014 - 4:15pm End : 02/26/2014 - 5:15pm

Category

Colloquium

Speaker

Lenny Fukshansky, Claremont McKenna College

Abstract

The celebrated Hilbert's 10th problem asks for an algorithm to decide whether a system of polynomial equations with integer coefficients has a nontrivial solution. A famous theorem of Matiyasevich (1970) states that no such algorithm exists in general. In fact, it is unlikely that such an algorithm exists even for a single polynomial of degree 4 or greater. On the other hand, algorithms are known to exist for systems of linear equations, as well as for a single quadratic equation. We will discuss a certain approach to the problem of searching for rational solutions of linear and quadratic equations, which also leads to an investigation of rational points on some related varieties over number fields and function fields. This approach involves height functions, which are common tools of modern arithmetic geometry.